Abstract
The existence of infinitely many nontrivial solutions for a nonlocal elliptic system of $(p_1,\ldots,p_n)$-Kirchhoff type with critical exponent is investigated. The approach is based on variational methods and critical point theory.
Highlights
For 0 < x < L, t ≥ 0 where, u = u(x, t) is the lateral displacement at the space coordinate x and the time t, E the Young modulus, ρ the mass density, h the crosssection area, L the length and ρ0 the initial axial tension, proposed by Kirchhoff as an extension of the classical D’Alembert’s wave equation for free vibrations of elastic strings
Kirchhoff-type boundary value problems model several physical and biological systems where, u describes a process which depend on the average of itself, as for example, the population density
Some early classical investigations of Kirchhoff equations can be seen in the papers [1,17,19,20,23,25,28,30] and the references therein
Summary
1 δ (c1) there is a global minimum of Φ which is a local minimum of Iλ, or (c2) there is a sequence of pairwise distinct critical points (local minima) of. |u|p⋆i dx pi/p⋆i is sequentially weakly lower semi-continuous in W01,pi(Ω). A special case of our main result is the following theorem. Has an unbounded sequence of pairwise distinct weak solutions. Our main result is the following theorem. Spn )[, the problem (1.1) has an unbounded sequence of weak solutions in E. It is well known that I is a Gateaux differentiable functional and sequentially weakly upper semi-continuous whose Gateaux derivative at the point u = N, by (M ) and Theorem 2.2 we get that J is sequentially weakly lower semi-continuous for ν ∈ [0, min(Sp1 , . By the assumption (M ), it is standard to see that Υλ ∈ C1(X, R), and a critical point of Υλ corresponds to a weak solution of problem (1.1). Ω sup(t1,...,tn)∈K(ξ) F (x, t1, . . . , tn)dx ξp for all k ∈ N
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